If `a , b , c ,` are in `A P ,a^2,b^2,c^2` are in HP, then prove that either `a=b=c` or `a , b ,-c/2` from a GP (2003, 4M)

Explanation

To solve the problem, we need to prove that if \( a, b, c \) are in Arithmetic Progression (AP) and \( a^2, b^2, c^2 \) are in Harmonic Progression (HP), then either \( a = b = c \) or \( a, b, -\frac{c}{2} \) form a Geometric Progression (GP).

### Step-by-Step Solution

1. **Understanding the Conditions**:
- Since \( a, b, c \) are in AP, we have:
\[
2b = a + c \quad \text{(1)}
...